Rank inequalities for the Heegaard Floer homology of Seifert homology spheres

Abstract

We establish three rank inequalities for the reduced flavor of Heegaard Floer homology of Seifert fibered integer homology spheres. Combining these inequalities with the known classifications of non-zero degree maps between Seifert fibered spaces, we prove that a map f : Y ′ → Y f:Y’ \to Y between Seifert homology spheres yields the inequality | deg ⁡ f | r a n k H F r e d ( Y ) ≤ r a n k H F r e d ( Y ′ ) |\deg f|\mathrm {rank} HF_{\mathrm {red}}(Y) \leq \mathrm {rank} HF_{\mathrm {red}}(Y’) . These inequalities are also applied in conjunction with an algorithm of Némethi to give a method to solve the botany problem for the Heegaard Floer homology of these manifolds.

Manolescu invariants of connected sums

“…We use graded roots, which were introduced by Némethi in order to study the Heegaard Floer homology of plumbed manifolds. The graded roots of Seifert spaces were studied in . Our brief introduction to graded roots will follow [, § 4] extremely closely.…”

“…Karakurt and Lidman define an abstract delta sequence as a pair with a well‐ordered finite set, and , with positive on the minimal element of . As we saw in § 7.1, an abstract delta sequence specifies a graded root up to a grading shift.…”

“…Theorem is proved using the technology of graded roots, introduced by Némethi , and refinements of the method of graded roots for Seifert spaces in . The proof is essentially borrowed from the partial calculation of for even by Hom, Karakurt, and Lidman .…”

Concordance maps in knot Floer homology

“…Note that this would imply [37, Conjecture 3.6] claiming that, if K K , then g(K) ≥ g(K ). Compare this with [18,Conjecture 9.4], which claims that if f : Y → Y is a non-zero degree map between integer homology spheres, then dim HF (Y ) ≥ dim HF (Y ). However, inequality (1.1) turns out to be false due to the following example constructed by Jennifer Hom.…”

Surgery Obstructions for Seifert Fibered Homology Spheres

“…Proposition 11. (Karakurt-Lidman, [14,Proposition 3.6]) Refinements and merges do not change H + (Γ) or H red (Γ).…”